1 CONTINUITY OF ALGEBRAIC OPERATIONS 81

(4.1.6) The mappings (x, y) -> sup(X y) and (x, y) -» inf(x, y) are uniformly
continuous in R x R.

As sup(x, y) = (x + y + \x - y\)/2 and inf(x, y) = (x + y - |* - ^|)/2,
the result follows from (4.1.1) and (3.20.9).

(4.1.7) All open intervals in R are homeomorphic to R.

From (4.1.1.) and (4.1.2) it follows that any linear function x -* ax + b,
with a T6 0, is a homeomorphism of R onto itself, for the inverse mapping
x~*a~lx — a"~lb has the same form. Any two bounded open intervals
]<*> /?[» ]y> <5[ are images of one another by a mapping x ~* ax + b, hence
are homeomorphic. Consider now the mapping x-+x/(l + \x\) of R onto
] —1, +1[; the inverse mapping is x -»x/(l — \x\) and both are continuous,
since x-* \x\ is. This proves R is homeomorphic to any bounded open
interval; finally, under the preceding homeomorphism of R onto ] — 1, +1[,
any unbounded open interval ]a, 4-oo[ or ] — oo, a[ of R is mapped onto a
bounded open interval contained in ] — 1, +1[, hence these intervals are also
homeomorphic to R.

(4.1.8) With respect to R x R, the function (x,y)-*x + y has a limit at
every point (a, b) ofR x R, except at the points (—00, + 00) flfld(+oo, —.00);
that limit is equal to + oo (resp. — oo) // one at least of the coordinates a, b
is +00 (resp. — oo).

Let us prove for instance that if a^ — oo, x + y has a limit equal to
+ 00 at the point (a, + oo). Given c e R, the relations x > b, y > c — b
imply x + y > c, and the intervals ]b, + oo] and ]c — b, + oo] are respectively
neighborhoods of a and + oo if b is taken finite and <a; hence our assertion.
The other cases are treated similarly.

(4.1.9) With respect to R x R, the function (x, y) -> xy has a limit at every
point (a, b) of S x 1, except at the points (0, +00), (0, — oo), (+00, 0),
(- oo, 0); that limit is equal to + oo (resp. — oo) if one at least of the coordinates
a, bis infinite, and if they have the same sign (resp. opposite signs).

Let us show for instance that if a > 0, xy has the limit + oo at the point
(a, + oo). Given c e R, the relations x > b, y > c/b, for b > 0, imply xy > c,
and the intervals ]b, + oo] and ]c/b, + oo] are neighborhoods of a and +00,
if b is taken finite and <a. Similar proofs for the other cases.ch that (x,y)eA and (y, z) e B}. Suppose